Eigenvalues, almost periodic functions, and the derivative of an integral. To facilitate this we give a new composition theorem of weighted pseudoalmost periodic functions. A function, f, from z to c is called almost periodic in bochner sense if and only if f is bounded and fnn. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Almostperiodic functions and functional equations springerlink. The concept was first studied by harald bohr and later generalized by vyacheslav stepanov, hermann weyl and abram samoilovitch besicovitch, amongst others. His interest was initially in finite dirichlet series. Almost periodic solution for a delayed lotkavolterra system. This paper is concerned with a singlespecies system with feedback control on time scales. You have remained in right site to begin getting this info. A key point is to build the analysis around the emerging theory of. Almostperiodic function encyclopedia of mathematics.
More precisely, a continuous function fx defined for all real values of 5 x is called almost periodic if for every. Newest almostperiodicfunction questions mathoverflow. Full text is available as a scanned copy of the original print version. Request pdf basic properties of almost periodic functions in section 2. The almost periodic functions form a natural example of a nonseparable normed space. Prouse, almostperiodic functions and functional equations, van nostrandreinhold, new york, ny, usa, 1971. Hierarchy of almostperiodic function spaces dipartimento di. Uniformly almost periodic functions and almost periodic. Note that we obtain the most important case if x is a banach space, and that the theory of almost periodic functions of real variable with values in a banach space, given by bochner 3, is in its essential lines similar to the theory of classical almost periodic functions. Pdf almost periodic functions, bohr compactification.
Bohr, who developed the notion of a uniformly almostperiodic function in his study of dirichlet series. When time scale or, our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Heteroclinic motions joining almost periodic solutions for. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a pdf plugin installed and enabled in.
Introduction finding a natural constructive treatment of the theory of almost periodic functions has long been a challenge for constructive mathematics, see 2, 5 and 7. In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, welldistributed almost periods. Pdf this paper is a survey of the basic properties of various type of almost periodic functions. Computing with almost periodic functions computing with almost periodic functions moody, r. Almost periodic solution for a delayed lotkavolterra. A first motivation for the study of almost periodic functions is the set of the various kinds of combining. For example, sin t has minimal period is 2 it follows from this that the minimal period for sin2t is the only exception is the constant function.
Pdf almost periodic functions, bohr compactification, and. Bohrs famous theory of almost periodic functions 4, if to arbitrary groups, and to show that it gives just the maximum range over which the fundamental results of fro. Composition of pseudo almost periodic functions and cauchy problems with operator of non dense domain. Periodic functions 2 now consider the functions y cos2t y sin2t the graphs of these functions are obtained from those of cos tand sin by compressing them along the xaxis by a factor of 2. A decomposition theorem is given showing that a function is in dollarlcub\cal paprcublcub\rm j\. As such, it has been a challenge for constructive mathematicians to find a natural treatment of them. Gibson, onthe almost periodicity oftrigonometric polynomials inconstructive mathematics, nederl.
By using the theory of exponential dichotomy on time scales and fixed point theory based on monotone operator, some simple conditions are obtained for the existence and uniqueness of positive almost periodic solution of the system. Almostperiodic function on a group encyclopedia of. Asymptotically almost periodic solution to a class of. Grande, hierarchy of almostperiodic function spaces, rendiconti di matematica, vol. Almostperiodic function on a group encyclopedia of mathematics. Every value of p 0 is a period and so it has no minimal period. Almost periodic function article about almost periodic. Elementary properties of almost periodic functions almost periodic functions are a generalization of the periodic functions which leave intact the property of completeness of fourier series. Ircub\sblcubarcub,xdollar if and only if it is the sum of an almost periodic function and an ergodic perturbation. Pseudo almost periodic functions and their applications. The theory of almost periodic functions was created and developed in its main features by bohr as a generalization of pure periodicity. Almost periodic functions on time scales and applications. Here, too, are a number of monographs on the subject, most notably by l. Download file pdf almost periodic type functions and ergodicity recognizing the quirk ways to acquire this books almost periodic type functions and ergodicity is additionally useful.
Most periodic functions have a minimal period, which is often called either the period or the base period. Almost periodic functions, bohr compactification, and differential equations article pdf available in milan journal of mathematics 661. Get a printable copy pdf file of the complete article 360k, or click on a page image below to browse page by page. The first half of the book lays the groundwork, noting the basic properties of almost periodic functions, while the second half of this work addresses applications whose main emphasis is on the solvability of ordinary or partial differential equations in the class of almost periodic functions. Brom, a constructive version of the fundamental theorem of the theory of almost periodic functions, thesis, university of california, san diego, 1974. Stepanov almost periodic functions edit the space s p of stepanov almost periodic functions for p. Get a printable copy pdf file of the complete article 284k, or click on a page image below to browse page by page. This paper is concerned with a delayed lotkavolterra system on time scales. Robust almost periodic dynamics for interval neural networks with mixed timevarying delays and discontinuous. Almost periodic function a function whose value is approximately repeated when its argument is increased by properly selected constants the almost periods. There are almost periodic functions which are not mean periodic. There are several inequivalent definitions of almost periodic functions.
Almost periodic solutions for a weakly dissipated hybrid system. Generalized asymptotically almost periodic and generalized asymptotically almost automorphic solutions of abstract multiterm fractional differential. Basic properties of almost periodic functions request pdf. Links to pubmed are also available for selected references. Composition theorems of stepanov almost periodic functions. Pdf a various types of almost periodic functions on banach. Asymptotically almost periodic solution to a class of volterra difference equations. Once bohr established his fundamental theorem, he was able to show that any continuous almost periodic function is the limit of a uniformly convergent sequence of trigonometric polynomials. An almostperiodic solution of hasegawawakatani equations with vanishing resistivity volume 146 issue 5 shintaro kondo skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. A childs garden of almost periodic functions caltech math. Almost periodic functions on locally compact groups.
Almostperiodic function article about almostperiodic. We also use the result for sequences to construct an almost periodic function whose. Because the circle is a compact abelian group, fourier theory may be used to approximate the periodic function f by finite sums of characters. Serial femtosecond crystallography on in vivogrown crystals drives elucidation of mosquitocidal cyt1aa bioactivation cascade. The most important examples are the trigonometric functions, which repeat over intervals of 2. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental. We study meromorphic functions in a strip almost periodic with respect to the spherical metric. Almost periodic functions, constructive mathematics, topological groups. Then we get a complete description of zeros and poles for this. In fact by truncating the series for the riemann zeta function. Almost periodic solutions for some convolution equations. If the inline pdf is not rendering correctly, you can download the pdf file here. Based on the theory of calculus on time scales, by using the properties of almost periodic functions and constructing a suitable lyapunov functional, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are obtained.
An ap function on the line 00 functions in the sense defined here. There are meanperiodic functions which are not almost periodic. A function fx in with complex values is called right almost periodic r. I was wondering whether a similar definition exists on the complex plane a locally. Asymptotically almost periodic solution to a class of volterra difference. A lmost periodic functions stepanov, weyl and besicov. The relation between a pseudo almost periodic function and its almost periodic component is discussed. Permanence and almost periodic solutions of a discrete ratiodependent leslie system with time delays and feedback controls yu, gang and lu, hongying, abstract and applied analysis, 2012.
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. Some properties of weighted pseudo almost periodic functions zheng, zheming, ding, huisheng, and nguerekata, gaston m. Get a printable copy pdf file of the complete article 365k, or click on a page image below to browse page by page. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. The theory of almost periodic functions was initiated by h. Almost periodic solutions for some convolution equations in. Grande, hierarchy of almost periodic function spaces, rendiconti di matematica, vol.
Full text full text is available as a scanned copy of the original print version. Bohrs famous theory of almost periodic functions 4, if to arbitrary groups, and to show that it gives just the maximum range over which the. Almost periodic sequences and functions with given values. Abstractin this paper, we first investigate some basic properties of remotely almost periodic functions and then show the existence and uniqueness of remotely almost periodic solutions to systems of differential equations with piecewise constant argument. Prouse, almost periodic functions and functional equations, van nostrandreinhold, new york, ny, usa, 1971. Robust almost periodic dynamics for interval neural networks with mixed time. In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. Almost periodicity is a structural property of functions, which is invariant with respect to the operations of addition and multiplication, and also in some cases with respect to division, differentiation, integration, and other limiting processes.
Based on the theory of calculus on time scales, by using the properties of almost periodic functions and constructing a suitable lyapunov functional, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are. The bohr almost periodic functions are essentially the same as continuous functions on the bohr compactification of the reals. Existence of almost periodic solutions of discrete time equations. S 41, marrakesh 2 department of mathematics, faculty of sciences semlalia, b. Eigenvalues, almost periodic functions, and the derivative of. Periodic functions 4 3 add the yaxis to match the phase shift, establishing the origin of the xaxis. In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably. Almost periodic solutions of a singlespecies system with. An ap function on the line 00 functions which are not mean periodic. I was wondering whether a similar definition exists on the complex plane a. The more vague and general notion of quasiperiodicity has even less to do with quasiperiodic functions in the mathematical sense.
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